Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilitySpringer, 31 dec. 2007 - 227 pagini Littlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense. It does so by letting us control certain oscillatory infinite series of functions in terms of infinite series of non-negative functions. Beginning in the 1980s, it was discovered that this control could be made much sharper than was previously suspected. The present book tries to give a gentle, well-motivated introduction to those discoveries, the methods behind them, their consequences, and some of their applications. |
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... ) < ∞ (1.1) then there exists a disjoint collection F ⊂ F such that every Q∈ F is contained in some Q∈ F. The proof is: For every Q∈ F, let Q be the maximal element (in the sense of set inclusion) of F that 2 1 Some Assumptions.
... disjoint dyadic cubes such that f = g + b, where g∞ ≤ 2dλ and b = ∑ Q∈F b(Q). Each function b(Q) has its support contained in Q and satisfies ∫ b(Q) dx = 0 |b(Q) |dx ≤ 2dλ|Q|. Moreover, the family F can be chosen so that ∑ F|Q ...
... (disjoint supports) with controlled L' norms, and which satisfy a cancelation condition, and with a total support that is also controlled, at least in terms of measure. Proof of Theorem 1.1. We will essentially prove the theorem twice ...
... disjoint family of dyadic cubes J'A such that b = XX, bø). where the functions bø satisfy the support and cancelation conditions, and also have | Wolars 2"Q/2)Q-2'No. Summing up the measures of the Q's, we get 2 2 |Q| < # / Alts #/ |f ...
... disjoint collection of dyadic intervals. Now split f into f1 + f2, where J f(a) – fj, if a € Jr.; no-' J £, (2.3) This definition forces f2 to equal JJ, if a € Jk: f2(a) = (#) if a £ U.J. This splitting has the consequence that, if I is ...
Cuprins
1 | |
9 | |
Exponential Square 39 | 38 |
Many Dimensions Smoothing | 69 |
The Calderón Reproducing Formula I | 85 |
The Calderón Reproducing Formula II | 101 |
The Calderón Reproducing Formula III | 129 |
Schrödinger Operators 145 | 144 |
Orlicz Spaces | 161 |
Goodbye to Goodλ | 189 |
A Fourier Multiplier Theorem | 197 |
VectorValued Inequalities | 203 |
Random Pointwise Errors | 213 |
References | 219 |
Index 223 | 222 |
Some Singular Integrals | 151 |
Alte ediții - Afișează-le pe toate
Weighted Littlewood-Paley Theory and Exponential-Square ..., Ediția 1924 Michael Wilson Previzualizare limitată - 2008 |